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This article is cited in 4 scientific papers (total in 4 papers) On the Zaremba problem for the A. G. Chechkinaab a Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, Russia b Institute of Mathematics with Computing Centre, Ufa Federal Research Centre of the Russian Academy of Sciences, Ufa, Russia
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     § 1. IntroductionThe classical problem of higher integrability of the gradients of solutions of elliptic equations dates back to [1], where linear divergence uniformly elliptic second-order equations with measurable coefficients on the plane were considered. Subsequently, in the multi-dimensional case higher integrability of the gradient of the solution to the Dirichlet problem in a domain with sufficiently regular boundary was established by Meyers in [2] for equations of the same form. Since then estimates for the higher integrability of the gradients of solutions are commonly called Meyers-type estimates. Zhikov [3] was the first to establish a Meyers-type estimate for the solution of the Dirichlet problem in a domain with Lipschitz boundary for the $p$-Laplace equation with variable exponent $p$ with logarithmic modulus of continuity. Then in [4] and [5] this result was strengthened and extended to systems of elliptic equations with variable integrability exponent. Note that in [3] the impetus for the study of Meyers estimates came from the thermistor problem, providing a joint description of the electric field potential and temperature (see [3], [6] and [7]). Systems of this kind also appear in the hydromechanics of quasi-Newtonian liquids. For Laplace’s equation, the mixed Zaremba problem, which was proposed by Wirtinger in a three-dimensional bounded domain with smooth boundary and inhomogeneous Dirichlet and Neumann conditions, was considered for the first time in [8]. The classical solvability of the problem was established using methods of potential theory under the assumption that the boundary of the open set on which the Neumann data are defined also satisfies certain smoothness conditions. The study of the properties of solutions to the Zaremba problem for elliptic second-order equation with variable regular coefficients dates back to [9], where, in particular, it was shown that solutions cease to be smooth at the interface of the Dirichlet and Neumann data. For second-order uniformly elliptic divergence equations with measurable coefficients, integral and pointwise estimates for solutions of the Zaremba problem were established in [10] under fairly general assumptions on the boundary of the domain. The problem of estimates for higher integrability of the gradient of the solution of the Zaremba problem is virtually unexplored. In this regard we can mention only [11], where the problem was formulated for a two-dimensional domain and the $p$-Laplace operator, with a rather particular assumption on the set on which the Dirichlet condition is defined (the paper itself used an approach of [12]), and also the recent papers [13]–[15], in which, for second-order linear elliptic equations, a Bojarski-Meyers estimate for the solution of the Zaremba problem was obtained, for a domain with Lipschitz boundary and a rapid change between Dirichlet and Neumann boundary conditions, with higher integrability exponent independent of the frequency of this change. In the present paper we consider, for the first time, the Zaremba problem for a nonlinear $p$-Laplace equation with most general conditions on the support of the Dirichlet data. Some results of this study were announced in [16]. Estimates of this kind, which are important in the theory of homogenization of problems with fast oscillation between boundary conditions, can be used for more accurate estimates for the convergence rate of the original solutions to that of the homogenized problem (for a similar problem in a domain perforated along the boundary, see [17]). § 2. Statement of the problemIn this paper we study the integral properties of weak solutions of the inhomogeneous $p$-elliptic equation with $p>1$ of the Zaremba problem in a bounded strictly Lipschitz domain $D\subset \mathbb{R}^n$, where $n>1$. To state the Zaremba problem we introduce the Sobolev function space $W^1_p(D,F)$. Here $F\subset \partial D$ is a closed set, $W^1_p(D,F)$ is the completion, in the norm of the space $W^1_p(D)$
$$
\begin{equation*}
\| u\|_{W^{1}_p(D,F)}=\biggl (\int_{D} |v|^p\,dx+\int_{D}|\nabla v|^p\,dx\biggr )^{1/p},
\end{equation*}
\notag
$$
of the class of functions infinitely differentiable on the closure of $D$ and which vanish near $F$. It is assumed a priori that the Friedrichs-type inequality holds for functions $v\in W^1_p(D,F)$; see below, and also see [18] and [19]. Assume that the matrix $A=(a_{ij})_{n\times n}$ with measurable entries is symmetric and uniformly positive definite, that is, $a_{ij}=a_{ji}$ and
$$
\begin{equation}
\alpha^{-1}|\xi|^2\leqslant \sum_{i,j=1}^na_{ij}(x)\xi_i\xi_j\leqslant\alpha |\xi|^2 \quad\text{for almost all }\ x\in D\ \text{ and all }\ \xi\in \mathbb{R}^n,
\end{equation}
\tag{2.2}
$$
where $\alpha$ is a positive constant. We set $G=\partial D\setminus F$ and consider the Zaremba problem of the following form for the $p$-elliptic equation
$$
\begin{equation}
\begin{cases} \mathcal{L}_p u:=\operatorname{div} (|\nabla u|^{p-2}A\nabla u)= l&\text{in } D, \\ u=0 &\text{on } F, \\ \dfrac{\partial u}{\partial \gamma }=0 &\text{on } G, \end{cases}
\end{equation}
\tag{2.3}
$$
where ${\partial u}/{\partial \gamma}$ is the outward conormal derivative of $u$, that is,
$$
\begin{equation*}
\frac{\partial u}{\partial \gamma}:=\sum_{i,j=1}^na_{ij}\frac{\partial u}{\partial x_j}\nu_i,
\end{equation*}
\notag
$$
$l$ is a linear functional in the dual space of $W^1_p(D,F)$, and $\nu$ is the unit outward normal vector to the boundary of the domain. By a solution to problem (2.3) we mean a function $u\in W^1_p(D,F)$ satisfying the integral identity
$$
\begin{equation}
\int_D|\nabla u|^{p-2}A\nabla u\cdot\nabla\varphi\,dx=-l(\varphi)
\end{equation}
\tag{2.4}
$$
for all test functions $\varphi\in W^1_p(D,F)$. In view of the Friedrichs inequality (2.1) the space $W^1_p(D,F)$ can be equipped with a norm involving only the gradient. In this case, with each element of the Sobolev space one can uniquely isometrically associate its gradient, that is, an element of $(L_p(D))^n$. Using the Hahn-Banach theorem (for example, as in § 1.1.14 of [18], in establishing the general form of a bounded linear functional in the dual space of a Sobolev space) one can easily show that the functional $l$ can be written as
$$
\begin{equation}
l(\varphi)= -\sum_{i=1}^n\int_D f_i\varphi_{x_i}\,dx,
\end{equation}
\tag{2.5}
$$
where $f_i\in L_{p'}(D)$, $p'=p/(p-1)$. Hence by (2.4), for each concrete functional, a solution to problem (2.3) is understood in the sense of the integral identity
$$
\begin{equation}
\int_D|\nabla u|^{p-2}A\nabla u\cdot\nabla\varphi\,dx=\int_D f\cdot\nabla\varphi\,dx
\end{equation}
\tag{2.6}
$$
for all test functions $\varphi\in W^1_p(D,F)$, in which the components of the vector function $f=(f_1,\dots,f_n)$ are $L_{p'}(D)$-functions. Note that the $p$-elliptic operator $\mathcal{L}_p$ acts from the space $W^1_p(D,F)$ to its dual space. To verify that the operator is monotone in $W^1_p(D,F)$ it suffices to show that the form
$$
\begin{equation*}
(|\xi|^{p-2}A\xi-|\zeta|^{p-2}A\zeta)\cdot (\xi-\zeta)>0 \quad \forall \,\xi,\zeta\in \mathbb{R}^n, \quad \xi\neq \zeta,
\end{equation*}
\notag
$$
is monotone. This holds if
$$
\begin{equation}
\max_{|\xi|=1}\frac{|A\xi|}{(A\xi, \xi)}<\frac{p}{|p-2|}
\end{equation}
\tag{2.7}
$$
(see [22], and also [21] for $p>2$). Now the $p$-elliptic operator $\mathcal{L}_p$ is monotone by condition (2.7). Remark 1. Using the machinery of the theory of monotone operators it can be shown that problem (2.3) is uniquely solvable in the Sobolev space of functions $W_p^1(D,F)$ (see, for example, Theorem 2.1 in [20], Ch. 2, § 2). Remark 2. Condition (2.7), which ensures the monotonicity of the operator $\mathcal{L}_p$, is required only for the unique solvability of the Zaremba problem. All other arguments in our paper depend only on the positive definiteness of the matrix $A$. The purpose of the present paper is to solve the problem of higher integrability of the gradient of the solution of (2.3) under the assumption $f\in L_{p'+\delta}(D)$, where $\delta>0$. The result that we obtain can be used to improve the available estimates for solutions of problems with rapidly changing type of boundary conditions. An important role in our analysis is played by the condition on the structure of the support $F$ of the Dirichlet data. To formulate the corresponding result, we recall that the capacity $C_q(K)$ of a compact set $K\subset \mathbb{R}^n$ for $1<q<n$ is defined by
$$
\begin{equation}
C_q(K)=\inf \biggl \{\int_{\mathbb{R}^n}|\nabla\varphi|^q\,dx\colon \varphi\in C^\infty_0 (\mathbb{R}^n),\, \varphi\geqslant 1 \ \text{on}\ K\biggr \}.
\end{equation}
\tag{2.8}
$$
Here the exponent $q$ is related to the exponent $p$ in (2.3) and the dimension $n$ of the space, and is defined as follows: if $p\in (1,n/(n-1)]$, then $q=(p+1)/2$, and if $p\in (n/(n-1),n]$, where $n>2$, then $q= np/(n+p)$. Note that for $n=2$ the second integral disappears. Below $B^{x_0}_r$ denotes the open ball of radius $r$ with centre $x_0$. Let us formulate our assumptions on the set $F$. A. If $1<p\leqslant n$, then we assume that, for each point $x_0\in F$, for $r\leqslant r_0$ we have
$$
\begin{equation}
C_q(F\cap \overline B^{\,x_0}_r)\geqslant c_0 r^{n-q},
\end{equation}
\tag{2.9}
$$
where the positive constant $c_0$ is independent of $x_0$ and $r$. B. If $p>n$, then the set $F$ is assumed to be nonempty. In each of these cases Friedrichs’s inequality (2.1) holds. Let us clarify this point in more details. Let $\mathcal{Q}_d$ be an open cube with edges of length $d$ parallel to coordinate axes. We assume that the Lipschitz domain $D$ has diameter $d$ and $D\subset \mathcal{Q}_d$. The capacity $C_p(K,\mathcal{Q}_{2d})$ of a compact set $K\subset\overline {\mathcal{Q}}_d$ relative to the cube $\mathcal{Q}_{2d}$ is defined by
$$
\begin{equation*}
C_p(K,\mathcal{Q}_{2d})=\inf \biggl \{\int_{\mathcal{Q}_{2d}}|\nabla\varphi|^p\,dx\colon \varphi\in C^\infty_0 (\mathcal{Q}_{2d}),\, \varphi\geqslant 1 \ \text{on}\ K\biggr \}.
\end{equation*}
\notag
$$
From the theorem in § 10.1.2 of [18] and the comments to Ch. 10 there (see the introduction to Ch. 10 of [18]) to the effect that the results established are also valid for Lipschitz domains it follows, in particular, that for $1<p\leqslant n$ any function $v\in W^1_p(D,F)$ satisfies Maz’ya’s inequality
$$
\begin{equation}
\int_D |v|^p\,dx\leqslant \frac{C(n,p,D)d^n}{C_p(F,\mathcal{Q}_{2d})}\int_D |\nabla v|^p\,dx
\end{equation}
\tag{2.10}
$$
with sharp constant on the right. Next we use condition (2.9) for $1<q<p\leqslant n$. First we note that by the definition of the capacity $C_p(K,\mathcal{Q}_{2d})$ and Hölder’s inequality we have the estimate
$$
\begin{equation}
C_q(K,\mathcal{Q}_{2d})\leqslant |\mathcal{Q}_{2d}|^{(p-q)/{p}}C_p^{q/p}(K,\mathcal{Q}_{2d}),
\end{equation}
\tag{2.11}
$$
in which $|\mathcal{Q}_{2d}|$ is the $n$-dimensional measure of the cube $\mathcal{Q}_{2d}$. Now we use the fact that for $1<q<n$ (see Proposition 4 in [23]) there exists a positive constant $\gamma(n,q)\geqslant 1$ such that
$$
\begin{equation*}
C_q(K)\leqslant C_q(K,\mathcal{Q}_{2d})\leqslant \gamma C_q(K).
\end{equation*}
\notag
$$
Hence from (2.11) and condition (2.9), in which $1<q<n$, we have $C_p(F,\mathcal{Q}_{2d})>0$. An appeal to (2.10) gives Friedrichs’s inequality (2.1). For $p>n$ we must use the definition of the inner (cubic) diameter of an open set (see [18], the end of § 10.2) and employ Theorem 1 in § 10.2.3 of [18], which implies (2.1). Let $\operatorname*{mes}_{n-1}(E)$ denote the $(n-1)$-dimensional Lebesgue measure of $E\subset\partial D$. Note that the condition
$$
\begin{equation}
\operatorname*{mes}_{n-1}(F\cap \overline B^{\,x_0}_r)\geqslant c_0r^{n-1},
\end{equation}
\tag{2.12}
$$
which is similar to (2.9), implies condition (2.9) itself. This follows from the estimate in Proposition 4 in § 9.1 of [18].
§ 3. The main resultRecall the definition of a Lipschitz domain $D$. Definition. We say that $D$ is a Lipschitz domain if for each point $x_0\in\partial D$ there exists an open cube $Q$ with centre $x_0$ such that its edges are parallel to coordinate axes, the edge length is independent of $x_0$, and in some Cartesian coordinate system with origin $x_0$ the set $Q\cap\partial D$ is the graph of a Lipschitz function $x_n=g(x_1, \dots, x_{n-1})$ whose Lipschitz constant is independent of $x_0$. We assume that the edge length of these cubes is $2R_0$, and the Lipschitz constant of the corresponding functions $g$ is $L$. For definiteness we assume that the set $Q\cap D$ lies above the graph of the function $g$. The main result of our paper is the following theorem, in which the constant $r_0$ in (2.9) is not larger than $R_0$. Theorem. If $f\in L_{p'+\delta_0}(D)$, where $\delta_0>0$ and $p'=p/(p-1)$, then there exists a positive constant $\delta(n,p,\delta_0)<\delta_0$ such that the solution of problem (2.3) satisfies Proof. First we find a higher integrability estimate for the gradient of the solution of (2.3) near the boundary of the domain $D$. This estimate is obtained by local straightening of the boundary $\partial D$.
At the first step we consider a local Cartesian coordinate system with origin $x_0$ such that the part of the boundary $\partial D$ that lies in the cube $\mathcal{Q}_{2R_0}$ is given in this system of coordinates by the equation $x_n=g(x')$, where $x'=(x_1,\dots,x_{n-1})$ and $g$ is an $L$-Lipschitz function. It is assumed that the domain $D_{R_0}=\mathcal{Q}_{2R_0} \cap D$ lies in the set of points where $x_n>g(x')$. Consider the new coordinate system in $\mathcal{Q}_{2R_0}$ defined by It is clear that the part of the boundary $\mathcal{Q}_{2R_0}\cap\partial D$ is transformed into a part of the hyperplane
$$
\begin{equation*}
P_{R_0}=\bigl\{y\colon |y_i|<R_0,\, i=1,\dots,n-1, \, y_n=0\bigr\}.
\end{equation*}
\notag
$$
At the first step we show that the image of the domain $\mathcal{Q}_{2R_0}$ contains the cube
$$
\begin{equation}
K_{R_0}=\bigl\{y\colon |y_i|<(1+\sqrt{n-1}\, L)^{-1}R_0, \, i=1,\dots,n\bigr\}.
\end{equation}
\tag{3.3}
$$
Indeed, if $y\in \widetilde {\mathcal{Q}}_{2R_0}$ and $|y_i|<\delta R_0$ for some $\delta\in (0,1)$ and $i=1,\dots,n-1$, then and since $g$ is Lipschitz continuous and $g(0)=0$, we have $|g(y')|\leqslant L|y'|<\sqrt{n-1}\, L\delta R_0$. Hence
$$
\begin{equation*}
\bigl(-R_0(1-\sqrt{n-1}\, L\delta), R_0(1-\sqrt{n-1}\, L\delta)\bigr)\subset \bigl(-R_0-g(y'), R_0-g(y')\bigr).
\end{equation*}
\notag
$$
If $\delta$ is chosen from the equality $\delta=1-\sqrt{n-1}\, L\delta$, then we have $K_{R_0}\subset \widetilde {\mathcal{Q}}_{2R_0}$.
At the first step of the proof, in the half-cube $K^+_{R_0}=K_{R_0}\cap \{y\colon y_n>0\}$, which lies in the image of the domain $D\cap \mathcal{Q}_{2R_0}$, problem (2.3) assumes the form
$$
\begin{equation}
\begin{cases} \mathcal{L}_1 u:=\operatorname{div}_y \bigl(|\nabla_y u-u_{y_n}\nabla_y g|^{p-2}\widetilde A(y)\nabla_y u\bigr)=\widetilde l &\text{in}\ K^+_{R_0}, \\ u=0 &\text{on}\ \widetilde F_{R_0}, \\ \dfrac{\partial u}{\partial \widetilde\gamma}=0 &\text{on} \ \widetilde G_{R_0} \end{cases}
\end{equation}
\tag{3.4}
$$
(the solution of this problem is denoted by the same letter). Here the matrix $\widetilde A(y)=(\widetilde a_{kl}(y))_{k,l=1}^n$ has the following entries. Let $k,l=1,\dots,n-1$. Then
$$
\begin{equation}
\begin{aligned} \, \widetilde a_{kl}(y) &= a_{kl}(y',y_n+g(y')), \\ \widetilde a_{nl}(y) &=\sum_{i=1}^{n-1}\frac{\partial g(y')}{\partial y_i} a_{il}(y',y_n+g(y'))+a_{nl}(y',y_n+g(y')), \\ \widetilde a_{kn}(y) &=\sum_{j=1}^{n-1}\frac{\partial g(y')}{\partial y_j} a_{kj}(y',y_n+g(y'))+a_{kn}(y',y_n+g(y')), \\ \widetilde a_{nn}(y) &=\sum_{i,j=1}^{n-1}\frac{\partial g(y')}{\partial y_i}\,\frac{\partial g(y')}{\partial y_j} a_{ij}(y',y_n+g(y')) \\ &\qquad+\sum_{j=1}^{n-1}\frac{\partial g(y')}{\partial y_j} a_{nj}(y',y_n+g(y') \\ &\qquad+ \sum_{i=1}^{n-1}\frac{\partial g(y')}{\partial y_i} a_{in}(y',y_n+g(y'))+ a_{nn}(y',y_n+g(y')). \end{aligned}
\end{equation}
\tag{3.5}
$$
It is easily seen that the symmetric matrix $\widetilde A(y)$ is uniformly positively definite, satisfies the condition
$$
\begin{equation}
\widetilde \alpha^{-1}|\xi|^2\leqslant \sum_{i,j=1}^n\widetilde a_{ij}(y)\xi_i\xi_j\leqslant\widetilde\alpha |\xi|^2 \quad \text{for almost all } \ y\in K^+_{R_0}\ \text{ and all } \ \xi\in \mathbb{R}^n
\end{equation}
\tag{3.6}
$$
with constant $\widetilde \alpha= \alpha(L+1)^2$, and the vector function $f$ involved in (2.5) is transformed into the vector function $\widetilde f=(\widetilde f_1(y),\dots, \widetilde f_n(y))$ with components
$$
\begin{equation}
\begin{aligned} \, \widetilde f_i(y) &= f_i(y',y_n+g(y')) \quad \text{for } i=1,\dots,n-1, \\ \widetilde f_n(y) &=\sum_{i=1}^{n-1}\frac{\partial g(y')}{\partial y_i} f_i(y',y_n+g(y'))+f_n(y',y_n+g(y')). \end{aligned}
\end{equation}
\tag{3.7}
$$
To verify this it suffices to change to the new variables in the integral identity (2.6) taking into account that the Jacobian of this change is 1, and then use the fact that $g$ is Lipschitz continuous. After this change the matrix is easily seen to be positive definite. We also obtain an explicit expression for the functions on the right-hand side of the transformed functional $\widetilde l$. For the sets $\widetilde F_{R_0}$ and $\widetilde G_{R_0}$ we have $\widetilde F_{R_0}=\widetilde F\cap P_{R_0}\cap K_{R_0}$ and $\widetilde G_{R_0}=\widetilde G\cap P_{R_0}\cap K_{R_0}$, where $\widetilde F$ and $\widetilde G$ are the images of the sets $F\cap \mathcal{Q}_{2R_0}$ and $G\cap \mathcal{Q}_{2R_0}$, respectively, and ${\partial u}/{\partial \widetilde\gamma}$ denotes the outward conormal derivative of the function $u$, which is generated by the matrix $\widetilde A$ and related to the operator in (3.4). Note that
$$
\begin{equation}
C_1(L)|\nabla_y u|\leqslant |\nabla_y u-u_{y_n}\nabla_y g|\leqslant C_2(L)|\nabla_y u|.
\end{equation}
\tag{3.8}
$$
Indeed, the upper estimate is clear, and the lower estimate (with positive constant) is obtained as follows. We have
$$
\begin{equation*}
\begin{aligned} \, &\sum_{j=1}^{n-1}\biggl(\frac{\partial u}{\partial y_j}- \frac{\partial g}{\partial y_j}\frac{\partial u}{\partial y_n}\biggr)^2+\biggl(\frac{\partial u}{\partial y_n}\biggr)^2 \\ &\qquad \geqslant \sum_{j=1}^{n-1}\biggl(\frac{\partial u}{\partial y_j}\biggr)^2+\biggl(1+\sum_{j=1}^{n-1}\biggl(\frac{\partial g}{\partial y_j}\biggr)^2\biggr)\biggl(\frac{\partial u}{\partial y_n}\biggr)^2-2\sum_{j=1}^{n-1} \biggl|\frac{\partial g}{\partial y_j}\frac{\partial u}{\partial y_j}\frac{\partial u}{\partial y_n}\biggr| \\ &\qquad \geqslant \sum_{j=1}^{n-1}\biggl(\frac{\partial u}{\partial y_j}\biggr)^2+\biggl(1+\sum_{j=1}^{n-1}\biggl(\frac{\partial g}{\partial y_j}\biggr)^2\biggr)\biggl(\frac{\partial u}{\partial y_n}\biggr)^2 \\ &\qquad\qquad- \frac1\varepsilon\sum_{j=1}^{n-1}\biggl(\frac{\partial g}{\partial y_j}\biggr)^2\biggl(\frac{\partial u}{\partial y_n}\biggr)^2-\varepsilon \sum_{j=1}^{n-1}\biggl(\frac{\partial u}{\partial y_j}\biggr)^2. \end{aligned}
\end{equation*}
\notag
$$
Choosing $\varepsilon=1/2$ and transferring the terms with coefficient $1/\varepsilon$ to the left-hand edge we obtain the required inequality with positive constant $C_1(L)$.
We extend the function $u$, which satisfies (3.4), as an even function relative to the hyperplane $\{y\colon y_n=0\}$. The extended function, which we also denote by $u$, satisfies
$$
\begin{equation}
\begin{cases} \mathcal{L}_2 u=\operatorname{div}(|\widetilde{\nabla}u|^{p-2}B(y)\nabla u)=l_h &\text{in}\ K_{R_0}\setminus \widetilde F_{R_0}, \\ u=0 &\text{on}\ \widetilde F_{R_0}. \end{cases}
\end{equation}
\tag{3.9}
$$
Here $\widetilde{\nabla}u$ coincides with $\nabla u-u_{y_n}\nabla g$ for $y_n>0$, and for $y_n<0$ it is also equal to ${\nabla u-u_{y_n}\nabla g}$, because the partial derivative $u_{y_n}$ extends as an odd function. The positive definite matrix $B(y)=\{b_{ij}(y)\}$ is such that the entries $b_{jn}(y)=b_{nj}(y)$ for $j\ne n$ are the odd extensions of the $\widetilde a_{jn}(y)$ in (3.4), and the remaining $b_{ij}(y)$ are the even extensions of the $\widetilde a_{ij}(y)$. It is easily seen that the matrix $B$ satisfies the ellipticity condition
$$
\begin{equation}
\beta^{-1}|\xi|^2\leqslant \sum_{i,j=1}^n b_{ij}(y)\xi_i\xi_j\leqslant\beta |\xi|^2 \quad\text{for almost all }\ y\in K_{R_0}\ \text{ and all } \ \xi\in \mathbb{R}^n,
\end{equation}
\tag{3.10}
$$
where $\beta$ depends on $\widetilde \alpha$. The components of the vector function $h=( h_1,\dots,h_n)$ in (3.9), which is involved in the representation of the functional $l_h$, are defined as follows: its components $h_i(y)$ for $i=1,\dots,n-1$ are the even extensions of the components $\widetilde f_i(y)$ in (3.4), and $h_n(y)$ is the odd extension of $\widetilde f_n(y)$. We also note that
$$
\begin{equation}
C_1(L)|\nabla u|\leqslant |\widetilde{\nabla}u|\leqslant C_2(L)|\nabla u|, \qquad y\in K_{R_0}.
\end{equation}
\tag{3.11}
$$
The solution of (3.9) is a function $u\in W^1_p(K_{R_0})$ satisfying the integral identity (see (2.6))
$$
\begin{equation}
\int_{K_{R_0}} |\widetilde{\nabla} u|^{p-2}B\nabla u\cdot\nabla\varphi\,dy=\int_ {K_{R_0}}h\cdot\nabla\varphi\,dy
\end{equation}
\tag{3.12}
$$
for all test functions $\varphi\in W^1_2(K_{R_0},\widetilde F_{R_0})$ lying in the completion, in the norm of the space $W^1_p(K_{R_0})$, of the set of infinitely differentiable functions defined on the closure of $K_{R_0}$ and vanishing near $\partial K_{R_0}$ and $\widetilde F_{R_0}$. The existence of a solution of (3.9) follows from the above transformations (see Remark 1).
Below we assume that
$$
\begin{equation*}
y_0\in K_{R_0/2}\setminus\partial K_{R_0/2}, \quad \text{where } R\leqslant\frac{1}{2}\operatorname{dist}(y_0,\partial K_{R_0/2}).
\end{equation*}
\notag
$$
We also write
$$
\begin{equation*}
{\,\rlap{-}\kern-1.5mm }\int_{\mathcal{Q}^{y_0}_{R}}f\, dx=\frac{1}{|\mathcal{Q}^{y_0}_R|}\int_{\mathcal{Q}^{y_0}_R}f\,dx,
\end{equation*}
\notag
$$
where $|\mathcal{Q}^{y_0}_R|$ is the $n$-dimensional measure of the cube $\mathcal{Q}^{y_0}_R$.
$\bullet$ First consider the case when $\mathcal{Q}^{y_0}_{3R}\subset K_{R_0}\setminus \widetilde F_{R_0} $. We substitute the test function $\varphi=(u-\lambda)\eta^p$ into the integral identity (3.12), where
$$
\begin{equation}
\lambda= {\,\rlap{-}\kern-1.5mm }\int_{\mathcal{Q}^{y_0}_{3R}} u\,dy,
\end{equation}
\tag{3.13}
$$
and the cut-off function $\eta\in C_0^{\infty}(\mathcal{Q}^{y_0}_{3R})$ is such that
$$
\begin{equation}
0<\eta\leqslant 1, \qquad \eta=1 \quad\text{in } \mathcal{Q}^{y_0}_{2R} \quad\text{and}\quad |\nabla \eta|\leqslant \frac CR.
\end{equation}
\tag{3.14}
$$
At the third step we show that the solution $u$ of problem (3.9) satisfies
$$
\begin{equation}
\int_{\mathcal{Q}^{y_0}_{2R}}|\nabla u|^p\, dy\leqslant C(n, p,\alpha, L)\biggl (\frac{1}{R^p}\int_{\mathcal{Q}^{y_0}_{3R}}|u-\lambda|^{p}\, dy+\int_{\mathcal{Q}^{y_0}_{3R}}|h|^{p'}\, dy \biggr).
\end{equation}
\tag{3.15}
$$
Indeed, taking the test function $\varphi=(u-\lambda)\eta^p$ in the integral identity (3.12), where $\eta$ is defined in (3.14), we obtain
$$
\begin{equation}
\begin{aligned} \, \notag &\int_{\mathcal{Q}^{y_0}_{3R}}|\widetilde \nabla u|^{p-2}B\nabla u\cdot \nabla u\,\eta^p\,dy =-p \int_{\mathcal{}Q^{y_0}_{3R}}\eta^{p-1}(u-\lambda)|\widetilde \nabla u|^{p-2}B\nabla u\cdot\nabla\eta\, dy \\ &\qquad\qquad +\int_{\mathcal{Q}^{y_0}_{3R}}\eta^p h\cdot \nabla u \,dy +p \int_{\mathcal{Q}^{y_0}_{3R}}\eta^{p-1}(u-\lambda) h\cdot \nabla \eta\, dy. \end{aligned}
\end{equation}
\tag{3.16}
$$
We have
$$
\begin{equation*}
C_1(L)\beta^{-1}|\nabla u|^{p-2}|\nabla u|^2\leqslant |\widetilde\nabla u|^{p-2}|B\nabla u\cdot\nabla u|\leqslant C_2(L)\beta |\nabla u|^{p-2}|\nabla u|^2
\end{equation*}
\notag
$$
(see (3.10) and (3.11)) and $0\leqslant \eta\leqslant 1$, and so by Young’s inequality
$$
\begin{equation}
\begin{aligned} \, p|\eta^{p-1}|u-\lambda|\,|\widetilde\nabla u|^{p-2}B\nabla u\cdot\nabla \eta| &\leqslant C(\beta, L) p\eta^{p-1}|u-\lambda|\,|\nabla u|^{p-2}|\nabla u|\, |\nabla \eta| \\ &\leqslant \varepsilon_1 |\nabla u|^p\eta^p+C(\varepsilon_1,\beta)|u-\lambda|^p|\nabla \eta|^p, \\ |\eta^p h\cdot \nabla u| &\leqslant \varepsilon_2 |\nabla u|^p\eta^p+C(\varepsilon_2)|h|^{p'}, \\ p|\eta^{p-1}|u-\lambda|h \cdot \nabla \eta| &\leqslant \varepsilon_3|h|^{p'}+C(\varepsilon_3, p)|u-\lambda|^p|\nabla \eta|^p. \end{aligned}
\end{equation}
\tag{3.17}
$$
Choosing $\varepsilon_1$, $\varepsilon_2$ and $\varepsilon_3$ appropriately, using equality (3.16) and inequality (3.17), and since the operator of problem (3.9) is elliptic, we have the estimate
$$
\begin{equation}
\int_{\mathcal{Q}^{y_0}_{3R}}|\nabla u|^p\eta^p\,dx\leqslant C\biggl(\int_{\mathcal{Q}^{y_0}_{3R}}|u-\lambda|^{p}|\nabla \eta|^{p}\,dx+\int_{\mathcal{Q}^{y_0}_{3R}} |h|^{p'}\,dx\biggr).
\end{equation}
\tag{3.18}
$$
Now inequality (3.15) follows since $\eta=1$ in $\mathcal{Q}^{y_0}_{2R}$ and $|\nabla\eta|\leqslant C/R$).
At the first step, using (3.15) and the Poincaré-Sobolev inequality
$$
\begin{equation*}
\biggl ({\,\rlap{-}\kern-1.5mm }\int_{\mathcal{Q}^{y_0}_{3R}}|u-\lambda|^{p}\,dx\biggr )^{1/p}\leqslant C(n,p)R\biggl ({\,\rlap{-}\kern-1.5mm }\int_{\mathcal{Q}^{y_0}_{3R}}|\nabla u|^q\,dx\biggr )^{1/q},
\end{equation*}
\notag
$$
where $q$ for $1<p\leqslant n$ is defined as in the paragraph following (2.8) and $q=(n+p)/2$) for $p>n$, we prove that
$$
\begin{equation}
\biggl({\,\rlap{-}\kern-1.5mm }\int_{\mathcal{Q}^{y_0}_{2R}}|\nabla u|^p\,dy\biggr )^{1/p}\leqslant C(n,p, \alpha,L)\biggl (\biggl ({\,\rlap{-}\kern-1.5mm }\int_{\mathcal{Q}^{y_0}_{4R}}|\nabla u|^q\,dy\biggr )^{1/q} +\biggl({\,\rlap{-}\kern-1.5mm }\int_{\mathcal{Q}^{y_0}_{4R}}|h|^{p'}\,dy\biggr )^{1/p}\biggr ).
\end{equation}
\tag{3.19}
$$
$\bullet$ Now consider the case when $\mathcal{Q}^{y_0}_{3R}\cap \widetilde F_{R_0}\ne \varnothing$. Substituting the test function $\varphi=u\eta^p$ into the integral identity (3.12) with the same cut-off function $\eta$ as above, we arrive at estimate (3.15) for $\lambda=0$. Using this estimate we find that
$$
\begin{equation}
\int_{\mathcal{Q}^{y_0}_{2R}}|\nabla u|^p\, dy\leqslant C(n, p, \alpha,L)\biggl (\frac{1}{R^p}\int_{\mathcal{Q}^{y_0}_{3R}}|u|^{p}\, dy+\int_{\mathcal{Q}^{y_0}_{3R}}|h|^{p'}\, dy \biggr).
\end{equation}
\tag{3.20}
$$
We estimate the first integral on the right in (3.20). Since $\mathcal{Q}^{y_0}_{3R}\cap \widetilde F_{R_0}\ne \varnothing$, there is a point $z_0\in \mathcal{Q}^{y_0}_{3R}\cap \widetilde F_{R_0}$ such that $\overline {\mathcal{Q}}^{\,z_0}_{R} \subset \overline {\mathcal{Q}}^{\,y_0}_{4R}$. We let $z\in F\cap \mathcal{Q}_{2R_0}$ denote the preimage of the set $z_0$ under the transformation (3.2). It is easily seen that the preimage of the closed cube $\overline {\mathcal{Q}}^{\,z_0}_{R}$ contains the closed ball $\overline B^z_{cR}$, where $c=c(L,n)>0$. If $1<p\leqslant n$ and condition (2.9) is met, then Maz’ya’s inequality holds:
$$
\begin{equation}
\biggl ({\,\rlap{-}\kern-1.5mm }\int_{\mathcal{Q}^{y_0}_{2R}}|u|^p\,dy\biggr )^{1/p}\leqslant C(n,p,c_0) R\biggl ({\,\rlap{-}\kern-1.5mm }\int_{\mathcal{Q}^{y_0}_{2R}}|\nabla u|^{q}\,dy\biggr )^{1/q}
\end{equation}
\tag{3.21}
$$
(see the theorem in § 10.1.2 of [18]). In this inequality the constant $q$ is as defined after (2.8).
If $p>n$, then by condition B in § 2 the set $\widetilde F_{R_0}$ is nonempty, and (3.21) holds for a constant $C$ independent of $c_0$. This can be verified using the definition of the inner (cubic) diameter of an open set (see [18], the end of § 10.2) and Theorem 1 in § 10.2.3 of [18]. So, from (3.20) and (3.21) we arrive at (3.19) again. Next we need the generalized Gehring lemma (see [24], and also Ch. VII in [25]). Let $g(x)$ and $f(x)$ be two nonnegative functions on $\mathcal{Q}_6$ such that $g\in L_p(\mathcal{Q}_6)$, $p>1$ and $f\in L_{p_0}(\mathcal{Q}_6)$, $p_0>p$. Lemma (generalized Gehring lemma). Assume that Using the above estimate, which holds for all cubes $\mathcal{Q}^{y_0}_{2R}$ under consideration, employing the generalized Gehring lemma, and recalling the definition of the edge length of the cube $K_{R_0}$ (see (3.3)), for $h\in L_{p'+\delta_0}(K_{R_0})$ where $\delta_0>0$, we have
$$
\begin{equation*}
\| \nabla u\|_{L_{p+\delta}(K_{R_0/4})}\leqslant C \bigl (\| \nabla u\|_{L_{p}(K_{R_0/2})}+ \| |h|^{p'/p}\|_{L_{p+\delta}(K_{R_0/2})}\bigr )
\end{equation*}
\notag
$$
for a positive constant $\delta=\delta (n,p,\delta_0)$ and for $C$ depending additionally on $R_0$. Since $u$ is an even function relative to the hyperplane $\{y\colon y_n=0\}$, this inequality can be written as
$$
\begin{equation}
\| \nabla u\|_{L_{p+\delta}(K^+_{R_0/4})}\leqslant C \bigl (\| \nabla u\|_{L_{p}(K^+_{R_0/2})}+ \| |\widetilde f|^{{p'}/{p}} \|_{L_{p+\delta}(K^+_{R_0/2})}\bigr)
\end{equation}
\tag{3.24}
$$
(see (3.4)). Using the inverse transform to (3.2), we note that the preimage of the half-cube $K^+_{R_0/2}$ is contained in the set $D_{R_0}$, and the preimage of the half-cube $K^+_{R_0/4}$ contains the set $D_{\theta R_0}$, where $\theta=\theta(n,L)>0$. Now an appeal to (3.7) and (3.24) shows that
$$
\begin{equation*}
\| \nabla u\|_{L_{p+\delta}(D_{\theta R_0})}\leqslant C \bigl (\| \nabla u\|_{L_{p}(D_{R_0})}+ \| |f|^{{p'}/{p}}\|_{L_{p+\delta}(D_{R_0})}\bigr ).
\end{equation*}
\notag
$$
Changing to the original Cartesian system of coordinates with origin at $x_0\in\partial D$, which we used from the very beginning, we obtain
$$
\begin{equation*}
\| \nabla u\|_{L_{p+\delta}(D\cap \mathcal{Q}^{x_0}_{\theta 2R_0})}\leqslant C \bigl (\| \nabla u\|_{L_{p}(D\cap \mathcal{Q}^{x_0}_{2R_0})}+ \| |f|^{{p'}/{p}}\|_{L_{p+\delta}(D\cap \mathcal{Q}^{x_0}_{2R_0})}\bigr).
\end{equation*}
\notag
$$
Since $x_0\in\partial D$ is an arbitrary boundary point and the boundary $\partial D$ is compact, there exists a finite covering of $\partial D$ such that the closed set
$$
\begin{equation*}
\mathcal{D}_{\theta_1 R_0}=\{x\in D\colon \operatorname{dist}(x,\partial D)\leqslant \theta_1 R_0\}, \qquad \theta_1=\theta_1(n,L)>0,
\end{equation*}
\notag
$$
is contained in the union of the sets $D\cap \mathcal{Q}^{x_i}_{\theta 2R_0}$, where $x_i\in\partial D$. Hence, using the triangle inequality and summing the inequalities
$$
\begin{equation*}
\| \nabla u\|_{L_{p+\delta}(D\cap \mathcal{Q}^{x_i}_{\theta 2R_0})}\leqslant C \bigl ( \| \nabla u\|_{L_{p}(D\cap \mathcal{Q}^{x_i}_{2R_0})}+ \| |f|^{p'/p}\|_{L_{p+\delta}(D\cap \mathcal{Q}^{x_i}_{2R_0})}\bigr ),
\end{equation*}
\notag
$$
we obtain the estimate
$$
\begin{equation*}
\| \nabla u\|_{L_{p+\delta}(\mathcal{D}_{\theta_1R_0})}\leqslant C \bigl (\| \nabla u\|_{L_{p}(D)}+ \| |f|^{p'/p}\|_{L_{p+\delta}(D)}\bigr ).
\end{equation*}
\notag
$$
The interior estimate
$$
\begin{equation}
\| \nabla u\|_{L_{p+\delta}(D\setminus \mathcal{D}_{\theta_1 R_0})}\leqslant C \bigl (\| \nabla u\|_{L_{p}(D)}+ \| |f|^{p'/p}\|_{L_{p+\delta}(D)}\bigr ),
\end{equation}
\tag{3.25}
$$
which is free from boundary conditions, can be proved much more simply. Indeed, let $D_\varpi\subset D$ be a strictly interior subdomain of $D$ lying at distance $\varpi$ from the boundary $\partial D$. We cover the closure $\overline{D_\varpi}$ by a finite number of cubes $\mathcal{Q}^i$ of edge length $\varpi/4$ and whose edges are parallel to coordinate axes. We also assume that this covering is such that an arbitrary closed cube with the same centre as $\mathcal{Q}^i$ and twice as great edge length lies in the domain $D$. Next we use the inequality (see (3.19))
$$
\begin{equation}
\biggl({\,\rlap{-}\kern-1.5mm }\int_{\mathcal{Q}^i_{\varpi/2}}|\nabla u|^p\,dy\biggr )^{1/p}\leqslant C \biggl (\biggl ({\,\rlap{-}\kern-1.5mm }\int_{\mathcal{Q}^{i}_{\varpi}}|\nabla u|^q\,dy\biggr )^{1/q} +\biggl({\,\rlap{-}\kern-1.5mm }\int_{\mathcal{Q}^{i}_{\varpi}}|f|^{p'}\,dy\biggr )^{1/p}\biggr ).
\end{equation}
\tag{3.26}
$$
Using the triangle inequality again and summing the above estimates with respect to $i$ for an appropriate choice of $\varpi$, we arrive at the interior estimate (3.25). As a result, combining the last two estimates and employing the energy inequality or
$$
\begin{equation*}
\int_D|\nabla u|^p\,dx\leqslant C \biggl(\int_D |f|^{p'(1+\delta/ p)}\,dx\biggr)^{p/(p+\delta)}
\end{equation*}
\notag
$$
for the first term on the right-hand sides of these estimates, we arrive at (3.1). This proves the theorem. § 4. Examples of the set $F$Example 1. Consider a set $F$ of zero $(n-1)$-dimensional measure which satisfies condition (2.9) (for a similar example, see [15]). For simplicity we consider a two-dimensional domain. For a justification of this example we introduce several auxiliary spaces. For $q\geqslant 1$ and $0<l\leqslant 1$ let the space $B^l_q$ be defined as the completion, with respect to the norm
$$
\begin{equation*}
\biggl (\int_{\mathbb{R}^n}\int_{\mathbb{R}^n}\vert \varphi(x+y)-2\varphi(x)+\varphi(x-y)\vert ^q \vert y\vert ^{-n-ql}\,dx\,dy\biggr )^{1/q}+\| \varphi\|_{L_q(\mathbb{R}^n)},
\end{equation*}
\notag
$$
of the set of compactly supported infinitely differentiable functions $\varphi\in C_0^\infty(\mathbb{R}^n)$. For $1<q<\infty$ and $l>0$ we also consider the Riesz potential space $h^l_q$ and the Bessel potential space $H^l_q$ defined as the completions of the class $\varphi\in C_0^\infty(\mathbb{R}^n)$ with respect to the norms
$$
\begin{equation*}
\| \varphi\|_{h^l_q}=\|(-\Delta)^{l/2}\varphi\|_{L_q(\mathbb{R}^n)}\quad\text{and} \quad \| \varphi\|_{H^l_q}=\|(-\Delta+1)^{l/2}\varphi\|_{L_q(\mathbb{R}^n)},
\end{equation*}
\notag
$$
respectively. Here $\Delta$ is the Laplace operator,
$$
\begin{equation*}
(-\Delta)^{l/2}=F^{-1}|\xi|^l F\quad\text{and} \quad (-\Delta+1)^{l/2}=F^{-1}(1+|\xi|^2)^{l/2} F,
\end{equation*}
\notag
$$
where $F\varphi(\xi)$ is the inverse Fourier transform,
$$
\begin{equation*}
F\varphi(\xi)=(2\pi)^{-n/2}\int_{\mathbb{R}^n}e^{i x\cdot\xi}\varphi(x)\,dx.
\end{equation*}
\notag
$$
It is known that (see Corollary 1 to Theorem 1 in Ch. 10 of [18])
$$
\begin{equation}
C_1(n,q)\|(-\Delta)^{1/2}\varphi\|_{L_q(\mathbb{R}^n)}\leqslant \|\nabla \varphi\|_{L_q(\mathbb{R}^n)}\leqslant C_2(n,q)\|(-\Delta)^{1/2}\varphi\|_{L_q(\mathbb{R}^n)}
\end{equation}
\tag{4.1}
$$
for each function $\varphi\in C^\infty_0(\mathbb{R}^n)$. For each of the function spaces $S^l_q=H^l_q$, $S^l_q=B^l_q$ and $S^l_q=h^l_q$ the capacity of a compact set $K\subset \mathbb{R}^n$ is defined by
$$
\begin{equation*}
\operatorname{cap}(K,S^l_q)=\inf\{\| u\|^q_{S^l_q}\colon \varphi\in C^\infty_0(\mathbb{R}^n),\ \varphi\geqslant 1 \text{ on } K\}.
\end{equation*}
\notag
$$
We will only be interested in the case when $0<l\leqslant 1$. The following relations between various capacities are known: (i) if $\operatorname{diam}(K)\leqslant 1$ and $ql<n$, then (see [26])
$$
\begin{equation}
\operatorname{cap}(K,H^l_q)\thicksim \operatorname{cap}(K,h^l_q);
\end{equation}
\tag{4.2}
$$
(ii) if $1<q<\infty$, then (see [27], Proposition 4.4.4)
$$
\begin{equation}
\operatorname{cap}(K,H^l_q)\thicksim \operatorname{cap}(K,B^l_q);
\end{equation}
\tag{4.3}
$$
(iii) if $K\subset \mathbb{R}^n$ and $1<q<\infty$, then (see [28])
$$
\begin{equation}
\operatorname{cap}(K,B^l_q(\mathbb{R}^n))\thicksim \operatorname{cap}(K,H^{l+1/q}_q(\mathbb{R}^{n+1})).
\end{equation}
\tag{4.4}
$$
From (4.2)–(4.4) it follows that if $K\subset \mathbb{R}^n$, $\operatorname{diam}(K)\leqslant 1$ and $ql<n$, then
$$
\begin{equation}
\operatorname{cap}(K,h^l_q(\mathbb{R}^{n+1}))\thicksim \operatorname{cap}(K,H^{l-1/q}_q(\mathbb{R}^{n})).
\end{equation}
\tag{4.5}
$$
Now we construct the required example on the plane for $n=1$, $l=1$ and $1<q<2$ in (4.5). In this case, if $K\subset \mathbb{R}$ and $\operatorname{diam}(K)\leqslant 1$, then
$$
\begin{equation*}
\operatorname{cap}(K,h^1_q(\mathbb{R}^{2}))\thicksim \operatorname{cap}(K,H^{1-1/q}_q(\mathbb{R})).
\end{equation*}
\notag
$$
By (4.1) we have the equivalence
$$
\begin{equation*}
\operatorname{cap}(K,h^1_q(\mathbb{R}^{2}))\thicksim C_q(K),
\end{equation*}
\notag
$$
where $C_q(K)$ is the above capacity of the compact set $K$. Hence
$$
\begin{equation}
C_q(K)\thicksim \operatorname{cap}(K,H^{1-1/q}_q(\mathbb{R})).
\end{equation}
\tag{4.6}
$$
Let $\{l_j\}$ be a decreasing sequence of positive numbers such that $2l_{j+1}<l_j$, ${j=1,2,\dots}$, and $\Delta_1$ be a closed interval of length $l_1\leqslant 1$ on the $Ox_1$-axis. Let ${E_1\subset \Delta_1}$ be the union of two closed intervals $\Delta_2$ and $\Delta_3$ of length $l_2$ adjoining the endpoints of the interval $\Delta_1$ (that is, from the closed interval $\Delta_1$ we remove the interval of length $l_1 - 2l_2$ with centre at the midpoint of $\Delta_1$). We proceed similarly with the intervals $\Delta_2$ and $\Delta_3$ (the role of $l_2$ is now played by $l_3$, that is, we remove the middle intervals of length $l_2 - 2l_3$). As a result, we obtain four closed intervals of length $l_3$. Let $E_2$ be the union of these intervals. Proceeding in this way we ultimately set $F=\bigcap_{j=1}^\infty E_j$. According to [29] (see also [30]), the assertions that and
$$
\begin{equation}
\sum_{j=1}^\infty 2^{{j}/(1-q)} l_j^{(2-q)/(1-q)}<\infty
\end{equation}
\tag{4.7}
$$
are equivalent. Thus, under condition (4.7) we have Consider the case of $q=(p+1)/{2}$. Now (4.7) takes the form
$$
\begin{equation*}
\sum_{j=1}^\infty 4^{j/(1-p)}l_j^{(3-p)/(1-p)}<\infty.
\end{equation*}
\notag
$$
Setting $l_j=a^{-j+1}$, where $a>2$, we have $2l_{l+1}<l_j$. Hence
$$
\begin{equation*}
\sum_{j=1}^\infty \biggl (\frac{1}{4}a^{3-p}\biggr)^{j/(p-1)} a^{(3-p)/(1-p)}<\infty.
\end{equation*}
\notag
$$
Since $1<p\leqslant 2$, this series converges for any $a$ in the interval $(2,4^{1/(3-p)})$. As a result, we obtain the Cantor set $F$, where at each step, from the closed interval $\Delta_1=[0,1]$ one removes intervals of length $(a-2)a^{-j}$, $j=1,2,\dots$ . It is well known that the one-dimensional Lebesgue measure of $F$ is zero. Indeed, at the $j$th step we remove $2^{j-1}$ intervals of length $(a-2)a^{-j}$, so that the sum of the lengths of all removed intervals is
$$
\begin{equation*}
\frac{a-2}{2}\sum_{j=1}^\infty\biggl(\frac{2}{a}\biggr)^j=1.
\end{equation*}
\notag
$$
In particular, for $a=3$ we obtain the classical Cantor set. Note that (see (4.8)) It remains to show that for an arbitrary point $x_0\in F$, for $r\leqslant r_0$ we have
$$
\begin{equation}
C_{(p+1)/2}(F\cap \overline B^{x_0}_r)\geqslant c_0 r^{(3-p)/{2}},
\end{equation}
\tag{4.10}
$$
where $B^{x_0}_r$ is the open disc of radius $r$ with centre $x_0$, and the positive constant $c_0$ is independent of $x_0$ and $r$. Recall that the capacity of the set $F^{x_0}_r=F\cap \overline B^{x_0}_r$ is defined by
$$
\begin{equation}
C_{(p+1)/2}(F^{x_0}_r)=\inf \biggl \{\int_{\mathbb{R}^2}|\nabla\varphi|^{(p+1)/{2}}\,dx\colon \varphi\in C^\infty_0 (\mathbb{R}^2),\, \varphi\geqslant 1 \ \text{on}\ F^{x_0}_r\biggr \}.
\end{equation}
\tag{4.11}
$$
It is clear that $F^{x_0}_r$ is the intersection of $F$ with the closed interval of length $r$ with centre $x_0$. If $r\leqslant r_0\leqslant a^{-1}$, then there exists a natural number $k_0$ such that $a^{-k_0-1}< r\leqslant a^{-k_0}$. Indeed, we have $x_0\in F$, and so, by the construction of the Cantor set, the point $x_0$ lies in the closed interval $I_{k_0}$ of length $a^{-k_0-2}$. By the choice of $a$ it easily follows that $I_{k_0}\cap F\subset F^{x_0}_r$ and, by the monotonicity of capacity,
$$
\begin{equation}
C_{(p+1)/2}(F^{x_0}_r)\geqslant C_{(p+1)/2}(I_{k_0}\cap F ).
\end{equation}
\tag{4.12}
$$
Applying the homothety
$$
\begin{equation}
y=\frac{x-x_0}{r}+x_0, \quad \text{where } r=a^{-k_0-2},
\end{equation}
\tag{4.13}
$$
to (4.11), by(4.12) we have
$$
\begin{equation}
C_{(p+1)/2}(F^{x_0}_r)\geqslant a^{-(k_0+2)(3-p)/2}C_{(p+1)/2}(\widetilde F_0) \geqslant a^{(p-3)/2}r^{(3-p)/{2}}C_{(p+1)/2}(\widetilde F_0),
\end{equation}
\tag{4.14}
$$
where $\widetilde F_0$ is the image of the set $I_{k_0}\cap F$. It remains to note that (4.13) transforms $\widetilde F_0$ into a shift of the Cantor set $F$ along the $Ox_1$-axis. Now the required relation (4.10) with constants $c_0=a^{(p-3)/2}$ and $r_0=1/a$ follows from (4.9) and (4.12). Example 2. This example (see [13]), which is related to fast oscillation between the homogeneous Dirichlet and Neumann data, has applications to homogenization theory. Let us divide the interval $I=[0,2]$ on the $x$-axis into equal interchanging closed intervals $\Delta_1^j$ and $\Delta_2^j$ of length $\varepsilon$, where $j=1,\dots, N$ and $\varepsilon={1}/{N}$. In this case $F$ is the union of the closed intervals $\Delta_1^j$. It is easily seen that condition (2.12) is met for $r_0\leqslant 1$. For an account of homogenization problems with rapid change between boundary conditions, see, for example, [31]–[33] and the references in [33]. 
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